Please show work and Free Body Diagram. For the points please do at least a and

Question

Please show work and Free Body Diagram.  For the points please do at least a and b.

A mass m= 2 kg is attached at the end of a uniform rigid bar of length L = 0.75 m and mass M= 1.0 kg to form an inverted pendulum, where a spring of stiffness k is attached at distance a = 0.25 m as shown in the figure. Assume small angletheta, Derive the differential equation of motion (show all details, including FDD). Determine the minimum value of the spring constant k for ensuring stability of the system. For k = 2.0 N/mm, determine the natural frequency in Hz. Using the result of part (c), if the bar is displaced 5degree and released, what is the maximum angular velocity attained by it?

Explanation / Answer

a)

I = mL^2 + ML^2 /3 = (m + M/3)*L^2

Distance of c.g from O = (mL + ML/2) / (m+M)

Balancing moments:

I*alpha = (m+M)g *(mL + ML/2) / (m+M) *Cos theta - k*a*theta

Writing alpha = theta_dotdot we get

(m + M/3)*L^2*theta_dotdot = (m+M)g *(mL + ML/2) / (m+M) *Cos theta - k*a*theta

Putting values,

1.3125*theta_dotdot = 18.39*Cos theta - k*0.25*theta

b)

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